Dynamic programming for finite ensembles of nanomagnetic particles

We use optimal control via a distributed exterior field to steer the dynamics of an ensemble of N interacting ferromagnetic particles which are immersed into a heat bath by minimizing a quadratic functional. Using the dynamic programming principle, we show the existence of a unique strong solution of the optimal control problem. By the Hopf–Cole transformation, the associated Hamilton–Jacobi–Bellman equation of the dynamic programming principle may be re-cast into a linear PDE on the manifold M=(S^2)^N, whose classical solution may be represented via Feynman–Kac formula. We use this probabilistic representation for Monte-Carlo simulations to illustrate optimal switching dynamics.